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Properties

Label 2-30e2-100.23-c1-0-44
Degree $2$
Conductor $900$
Sign $0.257 + 0.966i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.29 + 0.574i)2-s + (1.34 − 1.48i)4-s + (−0.0819 + 2.23i)5-s + (1.03 + 1.03i)7-s + (−0.880 + 2.68i)8-s + (−1.17 − 2.93i)10-s + (−3.35 − 4.61i)11-s + (−0.392 − 2.48i)13-s + (−1.92 − 0.742i)14-s + (−0.405 − 3.97i)16-s + (−4.26 − 2.17i)17-s + (−1.72 − 5.31i)19-s + (3.20 + 3.11i)20-s + (6.98 + 4.04i)22-s + (−0.0446 + 0.281i)23-s + ⋯
L(s)  = 1  + (−0.913 + 0.406i)2-s + (0.670 − 0.742i)4-s + (−0.0366 + 0.999i)5-s + (0.390 + 0.390i)7-s + (−0.311 + 0.950i)8-s + (−0.372 − 0.928i)10-s + (−1.01 − 1.39i)11-s + (−0.108 − 0.687i)13-s + (−0.515 − 0.198i)14-s + (−0.101 − 0.994i)16-s + (−1.03 − 0.526i)17-s + (−0.396 − 1.21i)19-s + (0.717 + 0.697i)20-s + (1.49 + 0.861i)22-s + (−0.00931 + 0.0587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.257 + 0.966i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (523, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.257 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.433980 - 0.333562i\)
\(L(\frac12)\) \(\approx\) \(0.433980 - 0.333562i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 - 0.574i)T \)
3 \( 1 \)
5 \( 1 + (0.0819 - 2.23i)T \)
good7 \( 1 + (-1.03 - 1.03i)T + 7iT^{2} \)
11 \( 1 + (3.35 + 4.61i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.392 + 2.48i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (4.26 + 2.17i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (1.72 + 5.31i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (0.0446 - 0.281i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-4.78 - 1.55i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3.26 + 1.06i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.62 - 0.573i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (0.181 + 0.131i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (0.707 - 0.707i)T - 43iT^{2} \)
47 \( 1 + (-5.53 + 2.82i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-10.4 + 5.33i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (10.3 + 7.49i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (8.77 - 6.37i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (0.542 - 1.06i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (11.1 + 3.62i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-14.6 - 2.31i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (0.564 - 1.73i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.71 + 1.38i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (7.48 + 10.3i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-2.49 - 4.90i)T + (-57.0 + 78.4i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.03887043438433012248920969545, −8.841106423123243983719115355241, −8.368177655839823117404613582837, −7.44430293289363665746008244514, −6.64845818409723494909796819122, −5.80903284508245664498018554358, −4.91724428471116668549867694589, −3.03404974045309563967262465685, −2.37582243701496533591578347884, −0.34348035800529817020507138980, 1.47112419020502664133575035516, 2.36466032649321911039482796948, 4.11624732244242985137989501628, 4.68944994792576083308603085023, 6.12211750758365481229737678658, 7.22853011924114150224831687261, 7.930157957166459642670264760701, 8.639275701470087896549856316930, 9.448193219805989782446432955571, 10.29040838535992999982353712426

Graph of the $Z$-function along the critical line